Uniqueness of the holomorphic Ahlfors–Hopf extremal on the disk

Let $\hat{h}:{\Bbb D}\to{\Bbb D}$ be a diffeomorphism of ${\Bbb D}$, a homeomorphism of $\overline{{\Bbb D}}$, and satisfy $\hat{h}|\partial{\Bbb D}=\operatorname{identity}$. Define

$$\hat{\Phi}={\Bbb K}(w,h)^{p-1}h_w\overline{h_{\overline{w}}}\,\eta(h).$$

Uniqueness conjecture. If $\hat{\Phi}$ is holomorphic, then $\hat{h}=\operatorname{identity}$.

This asserts uniqueness for the $L^p$-mean distortion problem on a finite-area Riemann surface in the homotopy class of the identity: the lifted extremal should be the identity whenever its Ahlfors–Hopf differential is holomorphic. The source presents this as an expected uniqueness statement, and no resolution is supplied here.